Toward a Strongly Polynomial Algorithm for the LinearComplementarity Problems with Sufficient Matrices

Komei Fukuda

ETH Zurich, Switzerland

fukuda@math.ethz.ch

https://www.inf.ethz.ch/personal/fukudak/

September 22, 2015

Linear Programming Problem (LP)

(GivenA ∈ Rm×d , b ∈ R

m, c ∈ Rd)

LP: max cT x = ∑dj=1 c j x j

subject to A x ≤ b ∑dj=1 ai j x j ≤ bi,∀i = 1, . . . ,m

x ≥ 0. x j ≥ 0,∀ j = 1, . . . ,d.

The set of feasible solutions{x : Ax ≤ b,x ≥ 0} is aconvexpolyhedron.

x3

x1

x2

x3

x1

x2

Linear Programming Problem (LP)

Linear programs withd = 3 andm = 3,20,50,100 constraints.

Three Algorithms for Linear Programming

Simplex Method

1

2

3

Criss-Cross Method

1

2

3

4

Interior-Point Method

All existing polynomial algorithms are based on interior-point mothods(or the ellipsoid method).

Our ultimate goal is to find a strongly polynomial algorithm.

Convex Quadratic Programming Problem (convex QP)

(GivenA ∈ Rm×d , G ∈ R

d×d: positive semidefinite, b ∈ Rm, c ∈ R

d)

QP: max cT x− 12 xT Gx

subject to A x ≤ b

x ≥ 0.

The convex QP (and the LP) admits a certificate for optimality.

Theorem [QP Duality Theorem, Dorn 1960]

If the QP has an optimal solution, then its dual QD:

QD: min bT y+ 12xT Gx

subject to Gx +AT y ≥ c

y ≥ 0

has an optimal solution and the optimal values are equal. Moreoverx

values of the optimal solutions can be taken to be the same.

Convex Quadratic Programming Problem (convex QP)

All algorithms solving the convex QP (and the LP) aim at finding this

certificate (i.e. primal and dual solutions).

Moreover, this optimality is equivalent to theKarush-Kuhn-Tucker

(KKT) conditions:

Primal Ax ≤ b , y ≥ 0 Dual

Feasibility x ≥ 0 , Gx +AT y ≥ c Feasibility

Aix = bi or yi = 0 ∀i Complementary

x j = 0 or (Gx +AT y) j = c j ∀ j Slackness

Linear Complementarity Problem with Sufficient Matrices (SU-LCP)

(Given a sufficient matrixM ∈ Rn×n, q ∈ R

n)

LCP: find two vectorsw,z ∈ Rn satisfying

w = M z+q,

w ≥ 0, z ≥ 0, and

wT z = 0.

The convex QP is a special case of SU-LCP, because the KT conditions

can be written as LCP with

M =

0 −A

AT G

, q =

b

−c

BecauseG is PSD, the matrixM is sufficient.

Sufficient Matrices and LCP

A matrix M is calledcolumnsufficient if

[ zi(Mz)i ≤ 0 for all i ] =⇒ [zi(Mz)i = 0 for all i] .

A matrix M is calledrow sufficient if MT is column sufficient, and

sufficient if both column and row sufficient.

• Many LP algorithms (e.g. simplex, criss-cross, interior-point) can be

generalized to solve sufficient-matrix LCPs (SU-LCPs).

• There is an interior-point algorithm that runs in time polynomial in

the size of input and one parameterκ (that is not bounded from

above), due to Kojima, Megiddo, Noma and Yoshise (1991).

• No algorithm is known to run in polynomial time.

• If SU-LCP is NP-hard, it implies NP=co-NP that is unlikely.

Murty’s LCP (1978)

M =

1 2 2 . . . 2 2

0 1 2 . . . 2 2

0 0 1 . . . 2 2...

0 0 0 . . . 1 2

0 0 0 . . . 0 1

, q =

−1

−1

−1...

−1

−1

.

• The criss-cross method (and Murty’s least index method) takes2n −1

pivots for Murty’s LCP.

• KF-Namiki (1994) showed that therandomizedcriss-cross takes

exactly (expected)n pivots.

Morris’s LCP (2002)

M =

1 2 0 . . . 0 0 0

0 1 2 . . . 0 0 0

0 0 1 . . . 0 0 0...

0 0 0 . . . 1 2 0

0 0 0 . . . 0 1 2

2 0 0 . . . 0 0 1

, q =

−1

−1

−1...

−1

−1

−1

.

• Morris (2002) proved that the randomized principal pivot method

takes at least((n−1)/2)! pivots on avarage for Morris’s LCP.

(The associated unique sink orientation is highly cyclic.)

• Foniok-KF-Gartner-Luthi (2008) showed that the criss-cross method

with any permutation of variables takes at mostO(n2) pivots.

Morris’s LCP for n = 3

(

0 0 0− − −

)

(

1 0 0+ − +

) (

0 1 0+ + −

) (

0 0 1− + +

)

(

1 1 0− + −

) (

1 0 1+ − −

) (

0 1 1− − +

)

(

1 1 1+ + +

)

12

3

K-Matrix LCP

A matrix isK-matrix if it is P-matrix and its off diagonal entries are

non-positive.

• This class of LCPs is well known to be very easy to solve.

• Foniok-KF-Gartner-Luthi (2008) showed that any principal pivot

method (including the criss-cross) starting at any basis takes at most

2n pivots starting from any complementary basis.

• Foniok-KF-Klaus (2010) gave a purely combinatorial proof of the

fast convergence in the setting of oriented matroids.

K-Matrices

A matrix isZ-matrix if its off diagonal entries are non-positive.

A matrix isK-matrix if it is both P-matrix and Z-matrix.

Theorem [Foniok-KF-Gartner-L uthi (2009)].The simple principal pivoting method withany pivot rule solves the LCPwith a K-matrix in at most2n pivots.

Namely, every directed path in any K-USO has length≤ 2n.

Theorem (K-Matrix Characterizations) [Fiedler-Pt ak (1962)].Let M be a Z-matrix. Then the following conditions are equivalent:

(a) ∃ x ≥ 0 such thatMx > 0;(b) ∃ x > 0 such thatMx > 0;

(c) M is non-singular andM−1 ≥ 0;(d) for eachx 6= 0, ∃ k such thatxk(Mx)k > 0;(e) M is a P-matrix, (and thus a K-matrix).

Part of the Original Proof

Translating Fiedler-Ptak Conditions into Vector Subspaces

Let M be anyn×n matrix. LetS = {1, . . . ,n} andT = {n+1, . . . ,2n}.

Consider the vector subspace

V := {x ∈ R2n : [I −M]x = 0}

wherex ∈V meansxS = MxT .

(a) ∃ x ≥ 0 such thatMx > 0⇐⇒ ∃x ∈V such thatxS > 0 andxT ≥ 0

(b) ∃ x > 0 such thatMx > 0⇐⇒ ∃x ∈V such thatx > 0

(c) M is non-singular andM−1 ≥ 0⇐⇒ x ∈V andxS ≥ 0 imply xT ≥ 0

(WhenM is non-singular,V = {x ∈ R2n : [M−1 −I]x = 0})

(c) M is non-singular andM−1 ≥ 0⇐⇒ y ∈V⊥ andyT ≥ 0 imply yS ≤ 0

(WhenM is non-singular,V⊥ = {y ∈ R2n : [I (M−1)T ]y = 0})

The Main Matrix Theorem

Theorem [Foniok-KF-Klaus (2010)].Let M be a Z-matrix withV = {x ∈ R

2n : [I −M]x = 0}.

Then, the following conditions are equivalent:

(a)∃x ∈V : xT ≥ 0 andxS > 0; (a*) ∃y ∈V⊥ : yS ≤ 0 andyT > 0;

(b) ∃x ∈V : x > 0; (b*) ∃y ∈V⊥ : yS < 0 andyT > 0;

(c) x ∈V andxS ≥ 0 =⇒ xT ≥ 0; (c*) y ∈V⊥ andyT ≥ 0 =⇒ yS ≤ 0;

(d) there is no s.r. circuitc ∈V (d*) there is no s.p. cocircuitc ∈V⊥.

(that is,M is a P-matrix);

Note: Acircuit is a nonzero vector with a minimal support. A vectorc is

s.r. (sign-reversing) ifci · cn+i ≤ 0 for all i = 1, . . . ,n.

The theorem above is a corollary to the OM Theorem where “matrix” is

replaced by “OM” and “V ” by “ V ”.

The Main OM Theorem

Theorem [Foniok-KF-Klaus (2010)].Let M = (S∪T,V ) be a Z-matroid.

Then, the following conditions are equivalent:

(a)∃X ∈ V : XT ≥ 0 andXS > 0; (a*) ∃Y ∈ V ∗ : YS ≤ 0 andYT > 0;

(b) ∃X ∈ V : X > 0; (b*) ∃Y ∈ V ∗ : YS < 0 andYT > 0;

(c) X ∈ V andXS ≥ 0 =⇒ XT ≥ 0; (c*) Y ∈ V ∗ andYT ≥ 0 =⇒ YS ≤ 0;

(d) there is no s.r. circuitC ∈ V (d*) there is no s.p. cocircuitD ∈ V ∗.

(that is,M is a P-matroid);

The proof is short and elementary.

First we prove (a)→ (b) → (c) → (d), by using the OM axioms.

The dual implications (a*)→ (b*) → (c*) → (d*) come for free (by

symbolic translation and duality). Finally, the implication (d)→ (a*)

follows from duality, and (d*)→ (a) comes for free.

A Consequence of The Main Theorem

The K-matroid characterization theorem can be used to give apurely

combinatorial proof of the fast convergence of K-LCP pivoting:

Theorem [Foniok-KF-Gartner-L uthi (2009)].The simple principal pivoting method withany pivot rule solves the LCP

with a K-matrix in at most2n pivots.

• Currently we are working on an extension of the K-matrix

characterizations to thehidden K-matrices.

• Eventually, we aim to show that the randomized criss-cross pivot rule

is (expected) strongly polynomial for LPs and convex QPs.

Conjecture (KF)

The Randomised Criss-Cross Method is an expected polynomial-time

algorithm for SU-LCP.

References

• J. Foniok, K. Fukuda, B. Gartner, and H.-J. Luthi. Pivoting in linear

complementarity: two polynomial-time cases.DiscreteComput.

Geom., 42:187–205, 2009. http://arxiv.org/abs/0807.1249.

• J. Foniok, K. Fukuda, and L. Klaus. Combinatorial characterizations

of K-matrices.LinearAlgebraandits Applications, 434:68–80,

2011. http://arxiv.org/abs/0911.2171.

• K. Fukuda and M. Namiki. On extremal behaviors of Murty’s least

index method.MathematicalProgramming, 64:365–370, 1994.

• K. Fukuda and T. Terlaky. Linear complementarity and oriented

matroids.Journalof theOperationsResearchSocietyof Japan,

35:45–61, 1992.

complementary dictionary

↓ ց

⊕·

r : top most r − ··⊕

stop↓ ց

s: left most s

r − + − ⊖ · · · ⊖

stop

ւ ց

p := max{r,s} q p q pq := min{r,s}

q q •

p • p • 0

diagonal pivotց ւ exchange pivot

complementary dictionary